A categorical account of the Metropolis-Hastings algorithm

TL;DR

This paper explores a categorical framework for the Metropolis-Hastings algorithm, enabling abstract reasoning about its properties like invariance and reversibility using advanced category theory concepts.

Contribution

It introduces a categorical formulation of MH, analyzing its core properties within Markov and CD categories, and establishes conditions for reversibility in this abstract setting.

Findings

Categorical formulation of MH invariance and reversibility

Synthetic conditions for MH reversibility with respect to target distributions

Framework for reasoning about probabilistic measures and kernels

Abstract

Metropolis-Hastings (MH) is a foundational Markov chain Monte Carlo (MCMC) algorithm. In this paper, we ask whether it is possible to formulate and analyse MH in terms of categorical probability, using a recent involutive framework for MH-type procedures as a concrete case study. We show how basic MCMC concepts such as invariance and reversibility can be formulated in Markov categories, and how one part of the MH kernel can be analysed using standard CD categories. To go further, we then study enrichments of CD categories over commutative monoids. This gives an expressive setting for reasoning abstractly about a range of important probabilistic concepts, including substochastic kernels, finite and $\sigma$-finite measures, absolute continuity, singular measures, and Lebesgue decompositions. Using these tools, we give synthetic necessary and sufficient conditions for a general MH-type sampler to be reversible with respect to a given target distribution.